Sorry for the long explanation, but it seems I just cannot avoid it...

I'll use your words as the basis for my answers here so I can try to be clear... You wrote, "Whether anyone has had exactly that experience is largely what's in dispute here" and to paraphrase,...What [is] the difference ... between the mystical experience, and a mere dwelling on the intuition or suspicion that there's a world out there that can't be grasped by our cognitive tools [?].

Here, which you seem to agree, is where the problem lies and I can only speak from my own perspective. The first caveat is that words are inadequate to explain the ineffable. This is a bridge that cannot be crossed by thought, and words represent thoughts.

In the midst of the absolute absence of self and of thought there is no thinking, philosophizing, musing, intuiting, or ruminating. It is only when not fully immersed in this state that explanations and thoughts come into play. By analogy, we can go on a raft ride down a river and be fully immersed in the experience and paying strict attention to every turn and danger and wonder of the experience. It is only after we are back at camp or at home that we muse over what occurred, how it felt, what it was like, etc. Similarly, when we are in the midst of extreme danger thought may completely disappear, including thought for the self; there is only reaction as if propelled and compelled by an invisible force, and oft times there is a sort of separation between the body and the observer that is just watching the scene unfold. HOWEVER, this still does not explain the realization of which I speak. The realization is that I am not the doer, not the body, not the mind, not the thoughts. Instead, I am something that is aware that is prior to, or behind, what occurs. This "something" is fundamental and without form.

Let's look at this definition of mysticism that the dictionaries seem to agree upon (maybe you have a better definition): "belief that union with or absorption into the Deity or the absolute, or the spiritual apprehension of knowledge inaccessible to the intellect, may be attained through contemplation and self-surrender.

vague or ill-defined religious or spiritual belief, especially as associated with a belief in the occult."

This definition is problematic in relation to what has often been called a mystical experience for at least three reasons:

First, an experience is impermanent; it comes and goes. What does not come and go is the realization of what it is that remains and always exists despite the movement and changes in life, thought, and experiences. There is a realization of this unchanging essence that I know. Instead of relating to the body and mind, I relate to that which is prior to the body and the mind, which is an ineffable space or capacity prior to, behind, before all movement, phenomena, thought, ideas, etc.

Second, the first word in the definition above is "belief," but the mystical experience (not an ideal word) as I know it is not a belief at all, but this word is used by those who cannot relate to what I described above. Belief means not to know something yet accept it as true anyway. To know what you are as the capacity has nothing to do with belief at all.

Third, the definition in question notes that there is a union or absorption into the Deity... This is not the point at all. Because one realizes that he/she IS the deity, god, life, consciousness, or whatever you want to call it, then the idea of "becoming" one with it is an absurdity, much like saying that you are becoming absorbed into your own body or that your toes are joining your feet.

In answer to the last paragraph you wrote: "The question I'd want to ask is what the difference is between the mystical experience, and a mere dwelling on the intuition or suspicion that there's a world out there that can't be grasped by our cognitive tools. I share that intuition, but I wouldn't call it a mystical perception of reality. Is that just a notational difference?":

The problem is that there is no explaining what is ineffable. There just are not any words, because words are metaphors, which are ideas borne of thought. At best, words can be pointers to guide one back to his own sense of self and what lies prior to it, the space out of which it arises. This may sound like doublespeak or an attempt to avoid offering an answer, but it is not. The closest analogy I can offer is qualia — there is really no way to convey or explain a feeling or what something feels like. The thinker, philosopher, academic, etc., offers answers from the outside-looking-in, but one who relates AS the capacity that contains — gives rise to — life and thought does so from the inside-out. To identify as the capacity for life instead of one of its contents is not intuition and it is not an intellectual understanding; it is a type of knowing or seeing.

Intuition is defined as the ability to understand something instinctively, without the need for conscious reasoning. But the ineffable essence of what I am, which is what lies at the foundation of what all else is, has nothing to do with understanding or reason.

I hope I have made my position clearer, but am concerned that I have not, because there is just no thinking one's way into the unexplainable, because all experiences, phenomena, and forms are explainable only by way of thought. The ineffable essence of what we are has no way of being defined, transferred, explained, or intuited.